Dynamic analysis of two-layer composite beams with partial interaction using a higher order beam theory
Introduction
Many advantages can be found in the composite structures over their isotropic counterparts, e.g. the steel–concrete composite structures possess a higher ratio of strength to weight than the conventional reinforced concrete structures. As a result, this type of structure has received a wide range of applications in civil engineering, aerospace, automotive, etc. Usually, the flexible shear studs are used in the interface to connect each sub-layer of composite beams, which may cause the interfacial slip between sub-layers. To study the slip effects, the first two-layer steel–concrete composite beam model, taking into account the partial interaction, was proposed by Newmark et al. [1] in 1951. In the model, each sub-layer was described by small deformation Euer–Bernoulli beam theory (EBT), which neglected the shear deformation throughout the beam body. This assumption, however, implies that the mechanical behavior may not be precisely predicted for deep composite beam structures [2] due to the significant shear effect; thus the necessity arises to refine their model.
Fairly recently, the Newark׳s model has been refined by a great deal of investigators [3], [4], [5], [6], [7], [8] to take into account the shear deformations using the Timoshenko beam kinematics. Either by finite element method (FEM) or analytical one, the linear static analyses of composite beams were performed by Refs. [6], [7], [8], [9]. For the dynamic problems, many studies can also be found to research the linear dynamic characteristcis [5], [10], [11] and nonlinear free vibration problem [12]. However, what has to be noted is that the shear correction factor introduced by Timoshenko beam theory (TBT) is attributed to the cross-sectional geometry of each sub-layer as well as the shearing stress around the section [13], i.e. this factor is no longer constant during the deformation for the sub-layers of composite beams, and it was also demonstrated by He and Yang [2]. To avoid the problem caused by the correction factor, higher order beam theory (HBT) has received much attention [13], [14], [15], whose kinematics is even more elaborate than that of the TBT. Using Reddy׳s [16] HBT, where a third order polynomial is taken to approximate the axial displacement of sub-layers, Chakrabarti et al. [14], [15] studied the static response of two-layer composite beams within linear and elastic range, and Chakrabarti et al. [13] extended it to the range of dynamics problems by FEM. Besides, Subramanian [17] developed a displacement based finite element for free vibration analysis of composite laminated beams, and outlined the analytical procedure for free vibration of beams using two types of HBTs; Li et al. [18] developed an exact finite element to conduct the free vibration analysis of laminated beams using hyperbolic shear deformation theory, and Vo and Thai [19] performed the static analysis of laminated beams by FEM based on various refined higher order deformation theories [20].
Most of the HBTs, including the Reddy׳s HBT, tended to neglect the transverse deformation of the beam, thereby, neglected the transverse normal strain and stress. To our point of view, capturing the transverse normal stress of sub-layers caused by the interfacial pressure or tension may reduce the gap from beam model to plane stress model. Trying to achieve this, Kant׳s [21], [22], [23] HBT is used in this study, where both the longitudinal and transverse higher order displacements are considered by approximating displacement in these two directions as third and second order polynomials, respectively. Subsequently, a dynamic model for two-layer composite beams, whose sub-layers follow Kant׳s HBT, is proposed by virtue of the principle of virtual work. And the finite element for the transient response and free vibration analyses is also developed. For the purpose of comparison, the finite element formulation for two-layer composite beam model based on TBT is also given. The improvement of the incorporated transverse deformation on the accuracy is examined through the comparisons among the results of composite beams using plane stress, Reddy׳s HBT and classical beam theories for the free vibration problem. In addition, the seismic analyses are carried out to investigate the performances of the present composite beam model on the transient response analysis. Finally, the responses of the composite beams to the moving load are also studied, especially to investigate the effects of the moving velocity and damping ratio of structure on the composite beam deflection.
Section snippets
Description of problems and assumptions
Let us consider a straight, planar, two-layer composite beam with possibly different cross-sections and materials including flexible shear connectors uniformly smeared over the interface. Sub-layers with overall span L, as is shown in Fig. 1(a), are marked with c and s. The layers are placed in Cartesian coordinate systems x–zc and x–zs, which originate from the centroid of each layer at the left end. And depths h1 and h2 are set to denote the centroid-interface distances of layers c and s,
Numerical example 1
Huang and Su [10] had analytically analyzed the free vibration characteristics of two-layer partial interaction composite beams according to the Newmark׳s hypothesis [1]. And Chakrabarti et al. [13] formulated each layer of composite beams with Reddy׳s HBT and carried out the free vibration analysis. In this section, the proposed higher order beam model and the corresponding finite element program implemented are verified through the comparisons with the results of free vibration characteristic
Conclusions
Kant׳s higher order beam theory, which takes into account both the transverse and axial higher order displacements, is applied to the dynamic problem of two-layer partial interaction composite beams. And the corresponding displacement based finite element is formulated, with which the free vibration analyses, analyses of transient response to the seismic excitation and moving load are carried out to verify the mathematical model and the program implemented and investigate the dynamic
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